I Can Photons travel faster than c?

I've looked up this question on the web, and I've gotten seeming conflicting answers.

According to Feynman's path integral - to find the probability of a photon being at A at time 1 and B at time 2 can be determined by taking an integral of the photon traveling over all possible paths. I understand that these paths more than anything are mathematical constructs and should not be taken super literally (since classical trajectories don't really make sense for quantum particles)

Regardless, I have seen on the internet that for points outside of the light cone, the integral results in very low probability of photons being detected, but still nonzero. This effect apparently becomes more significant for very small distances, and goes to zero at larger distances. This would mean there is an exponentially decaying probability of the photon having a speed greater than c. However, in other places I've seen people say pretty strictly that no it is not possible for the photon to have even slightly larger values of c. Even still, I've seen people say that while the photon may travel faster than c, information cannot. This I don't understand. If a photon is known to have been emitted at time 1, and absorbed at time 2, and this is faster than the speed of light - how did information not travel at the same speed? Though I know this has some implications on Causality..

No particle can travel faster than the speed of light in vacuum. I suspect that you are seeing a non vanishing probability density outside the light cone because you are not using the relativistic formulation of quantum mechanics. Are you solving the Schrödinger Eq., the Dirac Eq., or the Klein Gordon Eq. ?

No particle can travel faster than the speed of light in vacuum. I suspect that you are seeing a non vanishing probability density outside the light cone because you are not using the relativistic formulation of quantum mechanics. Are you solving the Schrödinger Eq., the Dirac Eq., or the Klein Gordon Eq. ?

Well I'm not solving anything - I was referring to what others have said about the matter. I was referring to the path integral in quantum field theory ( relativistic) for a photon.

If it's quantum field theory they are probably solving the Klein-Gordon equation. This equation is Lorentz invariant and should not allow faster than light solutions.

One thing to consider is that these paths are not really true particle paths. If they were then this would be classical mechanics and not quantum mechanics. It turns out however, that when you use the path integral formalism for a quantum particle you get the same probability density as you would if you use a propagator (the traditional approach). Some of these paths are classically forbidden, such as the faster than light paths you mention, but they are all required to sum to the correct probability amplitude.

Now all this assumes that your point "b" is in the light cone centered at point "a". If your end point "b" moves outside of this light cone then all of these paths should cancel each other out to give a total probability amplitude of zero.

If it's quantum field theory they are probably solving the Klein-Gordon equation. This equation is Lorentz invariant and should not allow faster than light solutions.

One thing to consider is that these paths are not really true particle paths. If they were then this would be classical mechanics and not quantum mechanics. It turns out however, that when you use the path integral formalism for a quantum particle you get the same probability density as you would if you use a propagator (the traditional approach). Some of these paths are classically forbidden, such as the faster than light paths you mention, but they are all required to sum to the correct probability amplitude.

Now all this assumes that your point "b" is in the light cone centered at point "a". If your end point "b" moves outside of this light cone then all of these paths should cancel each other out to give a total probability amplitude of zero.

I'm no expert in QM, but I always thought QFT was a separate animal, and left behind Schrodinger, Dirac, and Klein-Gordon equations for the "path integral formalism". But that is slightly digressing.

the summation due to classically forbidden terms are small, but the overall summation is still nonzero.

Right, this is what I meant by saying "they are all required to sum to the correct probability amplitude." When the particle moves from point a to b, it does not do so by following some path between the two points but instead it propagates as a field. The path integrals aren't necessarily the actual paths being followed by the particle so there is no reason to believe anything is travelling faster than light.

The important part is that if point b is outside of point a's light cone then the probability density at point b is zero even if the probabilities for individual paths are non-zero. You might think "hey if some paths have probabilities that are non-zero then do some paths need negative probabilities to get a total of zero at point b?" The answer is of course YES. The Klein-Gordon equation allows for negative probability densities, this is related to antiparticles.

Right, this is what I meant by saying "they are all required to sum to the correct probability amplitude." When the particle moves from point a to b, it does not do so by following some path between the two points but instead it propagates as a field. The path integrals aren't necessarily the actual paths being followed by the particle so there is no reason to believe anything is travelling faster than light.

The important part is that if point b is outside of point a's light cone then the probability density at point b is zero even if the probabilities for individual paths are non-zero. You might think "hey if some paths have probabilities that are non-zero then do some paths need negative probabilities to get a total of zero at point b?" The answer is of course YES. The Klein-Gordon equation allows for negative probability densities, this is related to antiparticles.

But the point he was trying to make was that the overall summation (sum of all the paths) was non-zero, so the probability density would be non-zero. He goes on to explain that even so this doesn't violate Causality.

I think the confusion may be coming from the fact that the path integral formulation used often relies on a WKB approximation. This means that the propagator may give non vanishing solutions outside the light cone where the approximation is not as good. The exact solution however will always be zero outside the light cone. Here is a link to a paper going into some detail on this. https://arxiv.org/pdf/gr-qc/9210019.pdf

I think the confusion may be coming from the fact that the path integral formulation used often relies on a WKB approximation. This means that the propagator may give non vanishing solutions outside the light cone where the approximation is not as good. The exact solution however will always be zero outside the light cone. Here is a link to a paper going into some detail on this. https://arxiv.org/pdf/gr-qc/9210019.pdf

Where? Please give a reference. And if it isn't a textbook or peer-reviewed paper, be prepared to be told that it isn't a valid reference and you should look at textbooks or peer-reviewed papers.

Well unfortunately I don't have any at the moment. The idea of what I was talking about is posted above in the Quora link, though. However, apparently Feynman said this exact same thing. That there is a probability of this occuring, however small, especially at small distances. He says: "The amplitudes for these possibilities are very small compared to the contribution from speed c; in fact, they cancel out when light travels over long distances."

"It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c. The amplitudes for these possibilities are very small compared to the contribution from speed c; in fact, they cancel out when light travels over long distances. However when distances are short... these other possibilities become vitally important and must be considered." Pg 89 QED

Implying at shorter distances, they wouldn't necessarily cancel out. Is this in contrast with what you said?

Staff: Mentor

This is an "I" level thread. That means you are assumed to have undergraduate level knowledge of the subject matter. Such knowledge is normally obtained by studying at least some textbooks or peer-reviewed papers. Have you studied any?

Staff: Mentor

The idea of what I was talking about is posted above in the Quora link, though.

Note carefully this statement from the Quora link:

"no measurement can affect any other measurement outside of its lightcone."

What that means is that, if we have two measurements that are spacelike separated, their results can't affect each other. So if one "measurement" is the emission of a photon, and the other "measurement" is the absorption of a photon, neither one can affect the other. That is why it is said that no information can be transmitted in this manner--information transmission requires the "source" to be able to affect the "receiver" in some controllable way.

Also, you have to be careful not to assume that the photon emitted at one event and the photon detected at the other event are "the same" photon. Photon's don't have little identity labels on them. You can say that a photon was emitted at event A and a photon was detected at event B, but you can't say the two are "the same" photon.

As good as this book is, it's still a pop science book, not a textbook or peer-reviewed paper. As such, it leaves out things. For example, it says that "these other possibilities are vitally important and must be considered", but it doesn't (IIRC) say exactly what they must be considered for.

"no measurement can affect any other measurement outside of its lightcone."

What that means is that, if we have two measurements that are spacelike separated, their results can't affect each other. So if one "measurement" is the emission of a photon, and the other "measurement" is the absorption of a photon, neither one can affect the other. That is why it is said that no information can be transmitted in this manner--information transmission requires the "source" to be able to affect the "receiver" in some controllable way.

Also, you have to be careful not to assume that the photon emitted at one event and the photon detected at the other event are "the same" photon. Photon's don't have little identity labels on them. You can say that a photon was emitted at event A and a photon was detected at event B, but you can't say the two are "the same" photon.

As good as this book is, it's still a pop science book, not a textbook or peer-reviewed paper. As such, it leaves out things. For example, it says that "these other possibilities are vitally important and must be considered", but it doesn't (IIRC) say exactly what they must be considered for.

Well alright. I have seen similar statements on forums such as PhysicsStack and while I agree it may not be a peer reviewed source it's just confusing hearing this seemingly conflicting statements - especially from the likes of somebody like Feynman. What is your take on the matter? Is a non-zero probability of detecting outside of the light cone nonsense? Or is something like what was said on Quora close to the truth - photons may travel faster than c, but will not violate causality? I just seem to be getting somewhat conflicting answers.

No. You have assumed that the photon emitted and the photon detected are "the same photon", which "travels" between the two events. You can't assume that. Go back and read my previous post again.

No, you're failing to recognize that you are making implicit assumptions which are false.

Well then how can we ever talk about the concept of photons moving at a particular speed, if it is not the time between the movement of nonzero probability? It doesn't matter to me if it's one photon or another, if the probability of photon being detected at point B after being emitted at point A at time X seconds, then that is good enough for the speed of a traveling photon?

Staff: Mentor

how can we ever talk about the concept of photons moving at a particular speed

In certain situations, this is a good enough approximation and it greatly simplifies the analysis. But it is only an approximation, and in many situations, such as the one under discussion here, it breaks down.

In certain situations, this is a good enough approximation and it greatly simplifies the analysis. But it is only an approximation, and in many situations, such as the one under discussion here, it breaks down.

No. See above.

Alright. I guess I'll just accept that I won't really understand the matter fully since I don't have much formal training in QFT. Thanks anyway.